Let be a probability measure on with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of . We show that almost every sample path of the random walk on , when realized in Culler and Vogtmann’s outer space, converges to the simplex of a free, arational tree. We then prove that the space of simplices of free and arational trees, equipped with the hitting measure, is the Poisson boundary of . Using Bestvina and Reynolds’s and Hamenstädt’s description of the Gromov boundary of the complex of free factors of , this gives a new proof of the fact, due to Calegari and Maher, that the realization in of almost every sample path of the random walk converges to a boundary point. We get in addition that , equipped with the hitting measure, is the Poisson boundary of .
"The Poisson boundary of ." Duke Math. J. 165 (2) 341 - 369, 1 February 2016. https://doi.org/10.1215/00127094-3166308